Calculation of Classical Fluctuation Path

As elsewhere discussed, see Volume I/Chapter 4 of the present five-volume work (Putz, 2016), the classical path for quantum fluctuation will not be written directly from the ordinary path free motion but using the similar result for harmonic motion upon which the free-harmonic condition at 0 will be imposed actually, the procedure is unfolded as follows. 

The time classical path is firstly considered for harmonic motion, see Volume I/Section 4.3.3 of the present five-volume work (Putz 2016)  

The real to imaginary time rotation is performed on the result (3.92) according with the Wick rule prescription of (3.36), being this equivalently of directly rewriting of expression (3.92) replacing the trigonometric 

Quantum Nanochemistry— Volume II Quantum Atoms and Periodicity 

The free-harmonic (to 0) limit is performed upon the expression (3.93) through employing the ordinary hyperbolic limit  

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